Non-orthogonal tensor diagonalization

Tichavský, Petr; Phan, Anh Huy; Cichocki, Andrzej

doi:10.1016/j.sigpro.2017.04.001

Cited by 15 publications

(12 citation statements)

References 28 publications

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“…For example, the JBD-OG/ORG method by H. Ghennioui et al [19], the JBD-LM method by O. Cherrak et al [6], the JBD-NCG method by D. Nion [26]. For more methods, we refer the readers to [12,5,33] and reference therein. A very useful matlab toolbox for tensor computationtensorlab [35], which is available at http://www.tensorlab.net, is also recommended for interested readers.…”

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confidence: 99%

An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

Cai¹,

Liu²

2017

SIAM J. Matrix Anal. & Appl.

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The exact/approximate non-orthogonal general joint block diagonalization (nogjbd) problem of a given real matrix set A = {A i } m i=1 is to find a nonsingular matrix W ∈ R n×n (diagonalizer) such that W T A i W for i = 1, 2, . . . , m are all exactly/approximately block diagonal matrices with the same diagonal block structure and with as many diagonal blocks as possible. In this paper, we show that a solution to the exact/approximate nogjbd problem can be obtained by finding the exact/approximate solutions to the system of linear equations A i Z = Z T A i for i = 1, . . . , m, followed by a block diagonalization of Z via similarity transformation. A necessary and sufficient condition for the equivalence of the solutions to the exact nogjbd problem is established. Two numerical methods are proposed to solve the nogjbd problem, and numerical examples are presented to show the merits of the proposed methods.Key words. joint block diagonalization, tensor decomposition, independent component analysis AMS subject classifications. 15A21,15A69, 65F30.1. Introduction. The joint block diagonalization problem, also called the simultaneous block diagonalization problem, is a particular case of the block term decomposition (BTD) of a third order tensor [10,11,14,26]. Such problem has found many applications in independent subspace analysis (e.g., [4,13,30,31]) and semidefinite programming (e.g., [18,8,2,9]). To specify the problem, we name the problem by nine capital letters wwxyyzzzz. The first two letters, ww, indicate the type of the matrices in the matrix set, sy/he for real symmetric/complex Hermitian matrices, ge for general matrices. The second letter, x, indicates that the problem is solved in the exact sense or the approximate sense, e for the former and a for the latter. The next two letters, yy, indicate the type of the diagonalizer, no/nu for non-orthogonal/non-unitary matrix, o/u for orthogonal/unitary matrix(often left as blank). The last four letters, zzzz, indicate the computation performed, jd for joint diagonalization, jbd for joint block diagonalization, gjbd for general joint block diagonalization. Next, we first give some definitions, then formulate the wweyyjbd problem and the wweyygjbd problem mathematically.Definition 1.1. We call τ n = (n 1 , . . . , n t ) a partition of positive integer n if n 1 , n 2 , . . . , n t are all positive integers and the sum of them is n, i.e., t i=1 n i = n. The integer t is called the cardinality of the partition τ n , denoted by card(τ n ). The set of all partitions of n is denoted by T n .Definition 1.2. Given a partition τ n = (n 1 , . . . , n t ), for any matrix A of order n, define its block diagonal part and off-block-diagonal part associated with τ n as Bdiag τn (A) = diag(A 11 , . . . , A tt ), OffBdiag τn (A) = A − Bdiag τn (A), respectively, where A ii is of order n i for i = 1, . . . , t. A matrix A is referred to as a τ n -block diagonal matrix if OffBdiag τn (A) = 0.

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confidence: 99%

An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

Cai¹,

Liu²

2017

SIAM J. Matrix Anal. & Appl.

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“…Based on Theorem , we can solve the exact GJBD problem by finding n linearly independent eigenvectors X of

P_{scriptA} false(λ false)

and then determining a permutation Π by revealing the block structure of Π T X H A i X Π. The clustering methods can find such a permutation; we will discuss the details in Section 4.…”

Section: Resultsmentioning

confidence: 99%

“…This stage is of great importance for determining the solutions to the GJBD problem; however, without knowing the number of the diagonal blocks, determining a correct τ n can be very tricky, especially when the noise is high and the block‐diagonal structure is fuzzy. From our numerical experience, the clustering method described in the work of Tichavsky et al is powerful and efficient for finding τ n and Π. Let

H = false[h_{i j} false] = true \sum_{i = 0}^{p} ((), false| X^{normalH} A_{i} X false| + false| X^{normalH} A_{i}^{normalH} X false|) .

…”

Section: Methodsmentioning

confidence: 99%

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

Cai

Cheng

Shi

2019

Numerical Linear Algebra App

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Summary In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set false{Aifalse}i=0p ( p ≥ 1), where a nonsingular matrix W (often referred to as a diagonalizer) needs to be found such that the matrices W HAiW 's are all exactly/approximately block‐diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by W = [x1,x2,…,xn]Π, where Π is a permutation matrix and xi's are eigenvectors of the matrix polynomial Pfalse(λfalse)=∑i=0pλiAi, satisfying that [x1,x2,…,xn] is nonsingular and where the geometric multiplicity of each λi corresponding with xi is equal to 1. In addition, the equivalence of all solutions to the exact GJBD problem is established. Moreover, a theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three‐stage method is proposed, and numerical results show the merits of the method.

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“…are different in some applications), the Tucker compression can be applied prior to the JD, 24 we would not consider such cases in this paper. The considered tensors all share the following latent common decomposition:…”

Section: Problem Formulationmentioning

confidence: 99%

A unitary joint diagonalization algorithm for nonsymmetric higher‐order tensors based on Givens‐like rotations

Miao

Cheng

et al. 2020

Numerical Linear Algebra App

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Based on Givens-like rotations, we present a unitary joint diagonalization algorithm for a set of nonsymmetric higher-order tensors. Each unitary rotation matrix only depends on one unknown parameter which can be analytically obtained in an independent way following a reasonable assumption and a complex derivative technique. It can serve for the canonical polyadic decomposition of a higher-order tensor with orthogonal factors. Furthermore, based on cross-high-order cumulants of observed signals, we show that the proposed algorithm can be applied to solve the joint blind source separation problem.The simulation results reveal that the proposed algorithm has a competitive performance compared with those of several existing related methods. K E Y W O R D Scanonical polyadic decomposition, higher-order tensor, joint blind source separation, joint diagonalization INTRODUCTIONIn the past few decades, there have been considerable works on tensor decompositions, 1-3 etc. Moreover, the joint diagonalization (JD) of a set of matrices or tensors has been recognized to be instrumental in signal processing, such as the blind source separation (BSS), image denoising, the independent component analysis (ICA), the multiple-input and multiple-output (MIMO) telecommunication systems, and so on. The JD of a set of matrices has a long history, and there are a huge number of related literatures, such as References 4-7, and the JD of a set of tensors can naturally be seen as its generalization. Tensor diagonalization and its link with the ICA were first introduced in Reference 8. A BSS algorithm by the simultaneous third-order tensor diagonalization was proposed in Reference 9, which shows that for third-order tensors, the computation of an elementary Jacobi-rotation is equivalent to the best rank-1 approximation. A generalization to the statistics of any order greater than two was done in Reference 10 but mainly considered the real number field. In addition, Maurandi and Moreau proposed one of the first coordinate algorithms for the nonorthogonal JD of any order complex tensors in Reference 11, which relies on a particular decomposition of diagonalizing matrices. In all the above works, factor matrices are assumed to be identical for all modes, which are only suitable for the problem of the single-set BSS. However, with the increasing availability of multiset and multimodal signals, the conventional BSS approaches encounter huge challenges, since they are inherently developed to process single-set data and, hence, without sufficiently considering this interset dependence. Therefore, joint BSS (JBSS) algorithms have attracted great interests, which can simultaneously recover the underlying multiple variables from multiple datasets. 7 In addition, the JBSS can keep the extracted components aligned across different datasets, 12 which is an important feature that is not provided by the single-set BSS methods.Numer Linear Algebra Appl. 2020;27:e2291.wileyonlinelibrary.com/journal/nla

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Non-orthogonal tensor diagonalization

Cited by 15 publications

References 28 publications

An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

An Algebraic Approach to Nonorthogonal General Joint Block Diagonalization

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

A unitary joint diagonalization algorithm for nonsymmetric higher‐order tensors based on Givens‐like rotations

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